Michael Barton | Penn State University (original) (raw)

Papers by Michael Barton

The Christian Century,(September 19, 1979), Sep 19, 1979

The television star I am thinking of wears expensive-looking, light-colored suits with dark ties,... more The television star I am thinking of wears expensive-looking, light-colored suits with dark ties, and faces a studio audience from behind a modern wooden desk flanked by plastic plants. A mural of skyscrapers is painted on the wall behind him. He has a boyish, handsome face, thick hair with an unemployed curl in front, and a great smile that you know survives off-camera. He begins the 90-minute show by chatting with his partner, and then proceeds to interview guests, flattering them with his trusting manner. Singers entertain. A ...

Pennsylvania History: A Journal of Mid-Atlantic Studies

Plymouth colony, Massachusetts, led off a score of inventive stud ies in family history; Anthony ... more Plymouth colony, Massachusetts, led off a score of inventive stud ies in family history; Anthony Wallace's Rockdale, an analysis of owners, workers, evangelical Christianity, and cotton manufactur ingin an antebellum Pennsylvania village, set the standard for thicklydetailed historical ethnography; Robert Gross's The Minutemen and Their World, a survey of Concord, Massachusetts, before and after the Revolution, showed that the new social history could meld artful narration with quantification; Paul Johnson's A ...

Numerical Methods for PDEs, 2018

This chapter studies the effect of the quadrature on the isogeometric analysis of the wave propag... more This chapter studies the effect of the quadrature on the isogeometric analysis of the wave propagation and structural vibration problems. The dispersion error of the isogeometric elements is minimized by optimally blending two standard Gauss-type quadrature rules. These blending rules approximate the inner products and increase the convergence rate by two extra orders when compared to those with fully-integrated inner products. To quantify the approximation errors, we generalize the Pythagorean eigenvalue error theorem of Strang and Fix. To reduce the computational cost, we further propose a two-point rule for C 1 quadratic isogeometric elements which produces equivalent inner products on uniform meshes and yet requires fewer quadrature points than the optimally-blended rules.

ACM Transactions on Graphics, 2021

CNC machining is the leading subtractive manufacturing technology. Although it is in use since de... more CNC machining is the leading subtractive manufacturing technology. Although it is in use since decades, it is far from fully solved and still a rich source for challenging problems in geometric computing. We demonstrate this at hand of 5-axis machining of freeform surfaces, where the degrees of freedom in selecting and moving the cutting tool allow one to adapt the tool motion optimally to the surface to be produced. We aim at a high-quality surface finish, thereby reducing the need for hard-to-control post-machining processes such as grinding and polishing. Our work is based on a careful geometric analysis of curvature-adapted machining via so-called second order line contact between tool and target surface. On the geometric side, this leads to a new continuous transition between "dual" classical results in surface theory concerning osculating circles of surface curves and osculating cones of tangentially circumscribed developable surfaces. Practically, it serves as an ef...

Computer-Aided Design, 2021

We investigate a recently introduced methodology for 5-axis flank computer numerically controlled... more We investigate a recently introduced methodology for 5-axis flank computer numerically controlled (CNC) machining, called double-flank milling [1]. We show that screw rotors are well suited for this manufacturing approach where the milling tool possesses tangential contact with the material block on two sides, yielding a more efficient variant of traditional flank milling. While the tool's motion is determined as a helical motion, the shape of the tool and its orientation with respect to the helical axis are unknowns in our optimization-based approach. We demonstrate our approach on several rotor benchmark examples where the pairs of envelopes of a custom-shaped tool meet high machining accuracy.

Computer Methods in Applied Mechanics and Engineering, 2018

We introduce a new approach to numerical quadrature on geometries defined by subdivision surfaces... more We introduce a new approach to numerical quadrature on geometries defined by subdivision surfaces based on quad meshes in the context of isogeometric analysis. Starting with a sparse control mesh, the subdivision process generates a sequence of finer and finer quad meshes that in the limit defines a smooth subdivision surface, which can be of any manifold topology. Traditional approaches to quadrature on such surfaces rely on per-quad integration, which is inefficient and typically also inaccurate near vertices where other than four quads meet. Instead, we explore the space of possible groupings of quads and identify the optimal macro-quads in terms of the number of quadrature points needed. We show that macro-quads consisting of quads from one or several consecutive levels of subdivision considerably reduce the cost of numerical integration. Our rules possess a tensor product structure and the underlying univariate rules are Gaussian, i.e., they require the minimum possible number of integration points in both univariate directions. The optimal quad groupings differ depending on the particular application. For instance, computing surface areas, volumes, or solving the Laplace problem lead to different spline spaces with specific structures in terms of degree and continuity. We show that in most cases the optimal groupings are quad-strips consisting of (1 × n) quads, while in some cases a special macro-quad spanning more than one subdivision level offers the most economical integration. Additionally, we extend existing results on exact integration of subdivision splines. This allows us to validate our approach by computing surface areas and volumes with known exact values. We demonstrate on several examples that our quadratures use fewer quadrature points than traditional quadratures. We illustrate our approach to subdivision spline quadrature on the well-known Catmull-Clark scheme based on bicubic splines, but our ideas apply also to subdivision schemes of arbitrary bidegree, including nonuniform and hierarchical variants. Specifically, we address the problems of computing areas and volumes of Catmull-Clark subdivision surfaces, as well as solving the Laplace and Poisson PDEs defined over planar unstructured quadrilateral meshes in the context of isogeometric analysis.

Precision Engineering, 2019

A new category of 5-axis flank computer numerically controlled (CNC) machining, called double-fla... more A new category of 5-axis flank computer numerically controlled (CNC) machining, called double-flank, is presented. Instead of using a predefined set of milling tools, we use the shape of the milling tool as a free parameter in our optimization-based approach and, for a given input free-form (NURBS) surface, compute a custom-shaped tool that admits highly-accurate machining. Aimed at curved narrow regions where the tool may have double tangential contact with the reference surface, like spiral bevel gears, the initial trajectory of the milling tool is estimated by fitting a ruled surface to the self-bisector of the reference surface. The shape of the tool and its motion then both undergo global optimization that seeks high approximation quality between the input free-form surface and its envelope approximation, fairness of the motion and the tool, and prevents overcutting. That is, our double-flank machining is meant for the semi-finishing stage and therefore the envelope of the motion is, by construction, penetration-free with the references surface. Our algorithm is validated by a commercial path-finding software and the prototype of the tool for a specific gear model is 3D printed.

Journal of Computational and Applied Mathematics, 2019

Calabrò et al. [9] changed the paradigm of the mass and stiffness computation from the traditiona... more Calabrò et al. [9] changed the paradigm of the mass and stiffness computation from the traditional element-wise assembly to a row-wise concept, showing that the latter one offers integration that may be orders of magnitude faster. Considering a B-spline basis function as a non-negative measure, each mass matrix row is integrated by its own quadrature rule with respect to that measure. Each rule is easy to compute as it leads to a linear system of equations, however, the quadrature rules are of the Newton-Cotes type, that is, they require a number of quadrature points that is equal to the dimension of the spline space. In this work, we propose weighted quadrature rules of Gaussian type which require the minimum number of quadrature points while guaranteeing exactness of integration with respect to the weight function. The weighted Gaussian rules arise as solutions of non-linear systems of equations. We derive rules for the mass and stiffness matrices for uniform C 1 quadratic and C 2 cubic isogeometric discretizations. Our rules further reduce the number of quadrature points by a factor of (p+1 2p+1) d when compared to [9], p being the polynomial degree and d the dimension of the problem, and consequently reduce the computational cost of the mass and stiffness matrix assembly by a similar factor.

The International Journal of Advanced Manufacturing Technology, 2018

A new method for 5-axis flank computer numerically controlled (CNC) machining using a predefined ... more A new method for 5-axis flank computer numerically controlled (CNC) machining using a predefined set of tappered ball-endmill tools (aka conical) cutters is proposed. The space of lines that admit tangential motion of an associated truncated cone along a general, doubly curved, free-form surface is explored. These lines serve as discrete positions of conical axes in 3D space. Spline surface fitting is used to generate a ruled surface that represents a single continuous sweep of a rigid conical milling tool. An optimization based approach is then applied to globally minimize the error between the design surface and the conical envelope. Our computer simulation are validated with physical experiments on two benchmark industrial datasets, reducing significantly the execution times while preserving or even reducing the milling error when compared to the state-of-the-art industrial software.

Computer Methods in Applied Mechanics and Engineering, 2018

We develop quadrature rules for the isogeometric analysis of wave propagation and structural vibr... more We develop quadrature rules for the isogeometric analysis of wave propagation and structural vibrations that minimize the discrete dispersion error of the approximation. The rules are optimal in the sense that they only require two quadrature points per element to minimize the dispersion error [8], and they are equivalent to the optimized blending rules we recently described. Our approach further simplifies the numerical integration: instead of blending two three-point standard quadrature rules, we construct directly a single two-point quadrature rule that reduces the dispersion error to the same order for uniform meshes with periodic boundary conditions. Also, we present a 2.5-point rule for both uniform and non-uniform meshes with arbitrary boundary conditions. Consequently, we reduce the computational cost by using the proposed quadrature rules. Various numerical examples demonstrate the performance of these quadrature rules.

Journal of Computational and Applied Mathematics, 2018

A numerical integration rule for multivariate cubic polynomials over n-dimensional simplices was ... more A numerical integration rule for multivariate cubic polynomials over n-dimensional simplices was designed by Hammer and Stroud [14]. The quadrature rule requires n + 2 quadrature points: the barycentre of the simplex and n + 1 points that lie on the connecting lines between the barycentre and the vertices of the simplex. In the planar case, this particular rule belongs to a two-parameter family of quadrature rules that admit exact integration of bivariate polynomials of total degree three over triangles. We prove that this rule is exact for a larger space, namely the C 1 cubic Clough-Tocher spline space over macro-triangles if and only if the split-point is the barycentre. This results into a factor of three reduction in the number of quadrature points needed to integrate the Clough-Tocher spline space exactly.

Metals, 2018

Manufacturing techniques that are applied to turbomachinery components represent a challenge in t... more Manufacturing techniques that are applied to turbomachinery components represent a challenge in the aeronautic sector. These components require high resistant super-alloys in order to satisfy the extreme working conditions they have to support during their useful life. Besides, in the particular case of Integrally Bladed Rotors (IBR), usually present complex geometries that need to be roughed and finished by milling and grinding processes, respectively. In order to improve their manufacturing processes, Super Abrasive Machining (SAM) is presented as a solution because it combines the advantages of the use of grinding tools with milling feed rates. However, this innovative technique usually needed high tool rotary speed and pure cutting oils cooling. These issues implied that SAM technique was not feasible in conventional machining centers. In this work, these matters were tackled and the possibility of using SAM in these five-axis centers with emulsion coolants was achieved. To verify this approach, Inconel 718 single blades with non-ruled surfaces were manufactured with Flank-SAM technique and conventional milling process, analyzing cutting forces, surface roughness, and dimension accuracy in both cases. The results show that SAM implies a suitable, controllable, and predictable process to improve the manufacture of aeronautical critical components, such as IBR.

Computer-Aided Design, 2017

We propose a new algorithm to detect patches of free-form surfaces that can be well approximated ... more We propose a new algorithm to detect patches of free-form surfaces that can be well approximated by envelopes of a rotational cone under a rigid body motion. These conical envelopes are a preferable choice from the manufacturing point of view as they are, by-definition, manufacturable by computer numerically controlled (CNC) machining using the efficient flank (peripheral) method with standard conical tools. Our geometric approach exploits multi-valued vector fields that consist of vectors in which the point-surface distance changes linearly. Integrating such vector fields gives rise to a family of integral curves, and, among them, linear segments that further serve as conical axes are quickly extracted. The lines that additionally admit tangential motion of the associated cone along the reference geometry form a set of candidate lines that are sequentially clustered and ordered to initialize motions of a rigid truncated cone. We validate our method by applying it on synthetic examples with exact envelopes, recovering correctly the exact solutions, and by testing it on several benchmark industrial datasets, detecting manufacturable conical envelope patches within fine tolerances.

Computer Aided Geometric Design, 2016

We investigate the close correspondence between barycentric coordinates and barycentric kernels f... more We investigate the close correspondence between barycentric coordinates and barycentric kernels from the point of view of the limit process when finer and finer polygons converge to a smooth convex domain. We show that any barycentric kernel is the limit of a set of barycentric coordinates and prove that the convergence rate is quadratic. Our convergence analysis extends naturally to barycentric interpolants and mappings induced by barycentric coordinates and kernels. We verify our theoretical convergence results numerically on several examples.

Computer-Aided Design, 2016

We introduce a new method that approximates free-form surfaces by envelopes of one-parameter moti... more We introduce a new method that approximates free-form surfaces by envelopes of one-parameter motions of surfaces of revolution. In the context of 5-axis computer numerically controlled (CNC) machining, we propose a flank machining methodology which is a preferable scallop-free scenario when the milling tool and the machined free-form surface meet tangentially along a smooth curve. We seek both an optimal shape of the milling tool as well as its optimal path in 3D space and propose an optimization based framework where these entities are the unknowns. We propose two initialization strategies where the first one requires a user's intervention only by setting the initial position of the milling tool while the second one enables to prescribe a preferable tool-path. We present several examples showing that the proposed method recovers exact envelopes, including semi-envelopes and incomplete data, and for general free-form objects it detects envelope sub-patches.

Prospects, 1982

The Historical study of the American character has been hobbled for several reasons, many of whic... more The Historical study of the American character has been hobbled for several reasons, many of which are summarized by David Stannard in “American Historians and the Idea of National Character: Some Problems and Prospects.” Stannard emphasizes that America has always been too complex a sociocultural system to have produced a uniform national character or a typical personality. He notes that cultural anthropologists have not found psychological uniformity even in small, preliterate communities. If scholars would study the variety of the nation's psychological characteristics instead—if they would search for the modal personality (most frequently occurring type) and the distribution of other personality types rather than only the basic personality type—then, at least in Stannard's opinion, they would avoid oversimplification, the most serious conceptual error. But even this more realistic approach retains methodological problems that are so serious that he suggests historians co...

Geochimica et Cosmochimica Acta, 1989

The Christian Century,(September 19, 1979), Sep 19, 1979

The television star I am thinking of wears expensive-looking, light-colored suits with dark ties,... more The television star I am thinking of wears expensive-looking, light-colored suits with dark ties, and faces a studio audience from behind a modern wooden desk flanked by plastic plants. A mural of skyscrapers is painted on the wall behind him. He has a boyish, handsome face, thick hair with an unemployed curl in front, and a great smile that you know survives off-camera. He begins the 90-minute show by chatting with his partner, and then proceeds to interview guests, flattering them with his trusting manner. Singers entertain. A ...

Pennsylvania History: A Journal of Mid-Atlantic Studies

Plymouth colony, Massachusetts, led off a score of inventive stud ies in family history; Anthony ... more Plymouth colony, Massachusetts, led off a score of inventive stud ies in family history; Anthony Wallace's Rockdale, an analysis of owners, workers, evangelical Christianity, and cotton manufactur ingin an antebellum Pennsylvania village, set the standard for thicklydetailed historical ethnography; Robert Gross's The Minutemen and Their World, a survey of Concord, Massachusetts, before and after the Revolution, showed that the new social history could meld artful narration with quantification; Paul Johnson's A ...

Numerical Methods for PDEs, 2018

This chapter studies the effect of the quadrature on the isogeometric analysis of the wave propag... more This chapter studies the effect of the quadrature on the isogeometric analysis of the wave propagation and structural vibration problems. The dispersion error of the isogeometric elements is minimized by optimally blending two standard Gauss-type quadrature rules. These blending rules approximate the inner products and increase the convergence rate by two extra orders when compared to those with fully-integrated inner products. To quantify the approximation errors, we generalize the Pythagorean eigenvalue error theorem of Strang and Fix. To reduce the computational cost, we further propose a two-point rule for C 1 quadratic isogeometric elements which produces equivalent inner products on uniform meshes and yet requires fewer quadrature points than the optimally-blended rules.

ACM Transactions on Graphics, 2021

CNC machining is the leading subtractive manufacturing technology. Although it is in use since de... more CNC machining is the leading subtractive manufacturing technology. Although it is in use since decades, it is far from fully solved and still a rich source for challenging problems in geometric computing. We demonstrate this at hand of 5-axis machining of freeform surfaces, where the degrees of freedom in selecting and moving the cutting tool allow one to adapt the tool motion optimally to the surface to be produced. We aim at a high-quality surface finish, thereby reducing the need for hard-to-control post-machining processes such as grinding and polishing. Our work is based on a careful geometric analysis of curvature-adapted machining via so-called second order line contact between tool and target surface. On the geometric side, this leads to a new continuous transition between "dual" classical results in surface theory concerning osculating circles of surface curves and osculating cones of tangentially circumscribed developable surfaces. Practically, it serves as an ef...

Computer-Aided Design, 2021

We investigate a recently introduced methodology for 5-axis flank computer numerically controlled... more We investigate a recently introduced methodology for 5-axis flank computer numerically controlled (CNC) machining, called double-flank milling [1]. We show that screw rotors are well suited for this manufacturing approach where the milling tool possesses tangential contact with the material block on two sides, yielding a more efficient variant of traditional flank milling. While the tool's motion is determined as a helical motion, the shape of the tool and its orientation with respect to the helical axis are unknowns in our optimization-based approach. We demonstrate our approach on several rotor benchmark examples where the pairs of envelopes of a custom-shaped tool meet high machining accuracy.

Computer Methods in Applied Mechanics and Engineering, 2018

We introduce a new approach to numerical quadrature on geometries defined by subdivision surfaces... more We introduce a new approach to numerical quadrature on geometries defined by subdivision surfaces based on quad meshes in the context of isogeometric analysis. Starting with a sparse control mesh, the subdivision process generates a sequence of finer and finer quad meshes that in the limit defines a smooth subdivision surface, which can be of any manifold topology. Traditional approaches to quadrature on such surfaces rely on per-quad integration, which is inefficient and typically also inaccurate near vertices where other than four quads meet. Instead, we explore the space of possible groupings of quads and identify the optimal macro-quads in terms of the number of quadrature points needed. We show that macro-quads consisting of quads from one or several consecutive levels of subdivision considerably reduce the cost of numerical integration. Our rules possess a tensor product structure and the underlying univariate rules are Gaussian, i.e., they require the minimum possible number of integration points in both univariate directions. The optimal quad groupings differ depending on the particular application. For instance, computing surface areas, volumes, or solving the Laplace problem lead to different spline spaces with specific structures in terms of degree and continuity. We show that in most cases the optimal groupings are quad-strips consisting of (1 × n) quads, while in some cases a special macro-quad spanning more than one subdivision level offers the most economical integration. Additionally, we extend existing results on exact integration of subdivision splines. This allows us to validate our approach by computing surface areas and volumes with known exact values. We demonstrate on several examples that our quadratures use fewer quadrature points than traditional quadratures. We illustrate our approach to subdivision spline quadrature on the well-known Catmull-Clark scheme based on bicubic splines, but our ideas apply also to subdivision schemes of arbitrary bidegree, including nonuniform and hierarchical variants. Specifically, we address the problems of computing areas and volumes of Catmull-Clark subdivision surfaces, as well as solving the Laplace and Poisson PDEs defined over planar unstructured quadrilateral meshes in the context of isogeometric analysis.

Precision Engineering, 2019

A new category of 5-axis flank computer numerically controlled (CNC) machining, called double-fla... more A new category of 5-axis flank computer numerically controlled (CNC) machining, called double-flank, is presented. Instead of using a predefined set of milling tools, we use the shape of the milling tool as a free parameter in our optimization-based approach and, for a given input free-form (NURBS) surface, compute a custom-shaped tool that admits highly-accurate machining. Aimed at curved narrow regions where the tool may have double tangential contact with the reference surface, like spiral bevel gears, the initial trajectory of the milling tool is estimated by fitting a ruled surface to the self-bisector of the reference surface. The shape of the tool and its motion then both undergo global optimization that seeks high approximation quality between the input free-form surface and its envelope approximation, fairness of the motion and the tool, and prevents overcutting. That is, our double-flank machining is meant for the semi-finishing stage and therefore the envelope of the motion is, by construction, penetration-free with the references surface. Our algorithm is validated by a commercial path-finding software and the prototype of the tool for a specific gear model is 3D printed.

Journal of Computational and Applied Mathematics, 2019

Calabrò et al. [9] changed the paradigm of the mass and stiffness computation from the traditiona... more Calabrò et al. [9] changed the paradigm of the mass and stiffness computation from the traditional element-wise assembly to a row-wise concept, showing that the latter one offers integration that may be orders of magnitude faster. Considering a B-spline basis function as a non-negative measure, each mass matrix row is integrated by its own quadrature rule with respect to that measure. Each rule is easy to compute as it leads to a linear system of equations, however, the quadrature rules are of the Newton-Cotes type, that is, they require a number of quadrature points that is equal to the dimension of the spline space. In this work, we propose weighted quadrature rules of Gaussian type which require the minimum number of quadrature points while guaranteeing exactness of integration with respect to the weight function. The weighted Gaussian rules arise as solutions of non-linear systems of equations. We derive rules for the mass and stiffness matrices for uniform C 1 quadratic and C 2 cubic isogeometric discretizations. Our rules further reduce the number of quadrature points by a factor of (p+1 2p+1) d when compared to [9], p being the polynomial degree and d the dimension of the problem, and consequently reduce the computational cost of the mass and stiffness matrix assembly by a similar factor.

The International Journal of Advanced Manufacturing Technology, 2018

A new method for 5-axis flank computer numerically controlled (CNC) machining using a predefined ... more A new method for 5-axis flank computer numerically controlled (CNC) machining using a predefined set of tappered ball-endmill tools (aka conical) cutters is proposed. The space of lines that admit tangential motion of an associated truncated cone along a general, doubly curved, free-form surface is explored. These lines serve as discrete positions of conical axes in 3D space. Spline surface fitting is used to generate a ruled surface that represents a single continuous sweep of a rigid conical milling tool. An optimization based approach is then applied to globally minimize the error between the design surface and the conical envelope. Our computer simulation are validated with physical experiments on two benchmark industrial datasets, reducing significantly the execution times while preserving or even reducing the milling error when compared to the state-of-the-art industrial software.

Computer Methods in Applied Mechanics and Engineering, 2018

We develop quadrature rules for the isogeometric analysis of wave propagation and structural vibr... more We develop quadrature rules for the isogeometric analysis of wave propagation and structural vibrations that minimize the discrete dispersion error of the approximation. The rules are optimal in the sense that they only require two quadrature points per element to minimize the dispersion error [8], and they are equivalent to the optimized blending rules we recently described. Our approach further simplifies the numerical integration: instead of blending two three-point standard quadrature rules, we construct directly a single two-point quadrature rule that reduces the dispersion error to the same order for uniform meshes with periodic boundary conditions. Also, we present a 2.5-point rule for both uniform and non-uniform meshes with arbitrary boundary conditions. Consequently, we reduce the computational cost by using the proposed quadrature rules. Various numerical examples demonstrate the performance of these quadrature rules.

Journal of Computational and Applied Mathematics, 2018

A numerical integration rule for multivariate cubic polynomials over n-dimensional simplices was ... more A numerical integration rule for multivariate cubic polynomials over n-dimensional simplices was designed by Hammer and Stroud [14]. The quadrature rule requires n + 2 quadrature points: the barycentre of the simplex and n + 1 points that lie on the connecting lines between the barycentre and the vertices of the simplex. In the planar case, this particular rule belongs to a two-parameter family of quadrature rules that admit exact integration of bivariate polynomials of total degree three over triangles. We prove that this rule is exact for a larger space, namely the C 1 cubic Clough-Tocher spline space over macro-triangles if and only if the split-point is the barycentre. This results into a factor of three reduction in the number of quadrature points needed to integrate the Clough-Tocher spline space exactly.

Metals, 2018

Manufacturing techniques that are applied to turbomachinery components represent a challenge in t... more Manufacturing techniques that are applied to turbomachinery components represent a challenge in the aeronautic sector. These components require high resistant super-alloys in order to satisfy the extreme working conditions they have to support during their useful life. Besides, in the particular case of Integrally Bladed Rotors (IBR), usually present complex geometries that need to be roughed and finished by milling and grinding processes, respectively. In order to improve their manufacturing processes, Super Abrasive Machining (SAM) is presented as a solution because it combines the advantages of the use of grinding tools with milling feed rates. However, this innovative technique usually needed high tool rotary speed and pure cutting oils cooling. These issues implied that SAM technique was not feasible in conventional machining centers. In this work, these matters were tackled and the possibility of using SAM in these five-axis centers with emulsion coolants was achieved. To verify this approach, Inconel 718 single blades with non-ruled surfaces were manufactured with Flank-SAM technique and conventional milling process, analyzing cutting forces, surface roughness, and dimension accuracy in both cases. The results show that SAM implies a suitable, controllable, and predictable process to improve the manufacture of aeronautical critical components, such as IBR.

Computer-Aided Design, 2017

We propose a new algorithm to detect patches of free-form surfaces that can be well approximated ... more We propose a new algorithm to detect patches of free-form surfaces that can be well approximated by envelopes of a rotational cone under a rigid body motion. These conical envelopes are a preferable choice from the manufacturing point of view as they are, by-definition, manufacturable by computer numerically controlled (CNC) machining using the efficient flank (peripheral) method with standard conical tools. Our geometric approach exploits multi-valued vector fields that consist of vectors in which the point-surface distance changes linearly. Integrating such vector fields gives rise to a family of integral curves, and, among them, linear segments that further serve as conical axes are quickly extracted. The lines that additionally admit tangential motion of the associated cone along the reference geometry form a set of candidate lines that are sequentially clustered and ordered to initialize motions of a rigid truncated cone. We validate our method by applying it on synthetic examples with exact envelopes, recovering correctly the exact solutions, and by testing it on several benchmark industrial datasets, detecting manufacturable conical envelope patches within fine tolerances.

Computer Aided Geometric Design, 2016

We investigate the close correspondence between barycentric coordinates and barycentric kernels f... more We investigate the close correspondence between barycentric coordinates and barycentric kernels from the point of view of the limit process when finer and finer polygons converge to a smooth convex domain. We show that any barycentric kernel is the limit of a set of barycentric coordinates and prove that the convergence rate is quadratic. Our convergence analysis extends naturally to barycentric interpolants and mappings induced by barycentric coordinates and kernels. We verify our theoretical convergence results numerically on several examples.

Computer-Aided Design, 2016

We introduce a new method that approximates free-form surfaces by envelopes of one-parameter moti... more We introduce a new method that approximates free-form surfaces by envelopes of one-parameter motions of surfaces of revolution. In the context of 5-axis computer numerically controlled (CNC) machining, we propose a flank machining methodology which is a preferable scallop-free scenario when the milling tool and the machined free-form surface meet tangentially along a smooth curve. We seek both an optimal shape of the milling tool as well as its optimal path in 3D space and propose an optimization based framework where these entities are the unknowns. We propose two initialization strategies where the first one requires a user's intervention only by setting the initial position of the milling tool while the second one enables to prescribe a preferable tool-path. We present several examples showing that the proposed method recovers exact envelopes, including semi-envelopes and incomplete data, and for general free-form objects it detects envelope sub-patches.

Prospects, 1982

The Historical study of the American character has been hobbled for several reasons, many of whic... more The Historical study of the American character has been hobbled for several reasons, many of which are summarized by David Stannard in “American Historians and the Idea of National Character: Some Problems and Prospects.” Stannard emphasizes that America has always been too complex a sociocultural system to have produced a uniform national character or a typical personality. He notes that cultural anthropologists have not found psychological uniformity even in small, preliterate communities. If scholars would study the variety of the nation's psychological characteristics instead—if they would search for the modal personality (most frequently occurring type) and the distribution of other personality types rather than only the basic personality type—then, at least in Stannard's opinion, they would avoid oversimplification, the most serious conceptual error. But even this more realistic approach retains methodological problems that are so serious that he suggests historians co...

Geochimica et Cosmochimica Acta, 1989